Set of Submagmas of Magma under Subset Relation forms Complete Lattice

Theorem

Let $\struct {A, \odot}$ be a magma.

Let $\SS$ be the set of submagmas of $A$.

Then:

the ordered set $\struct {\SS, \subseteq}$ is a complete lattice

where for every subset $\TT$ of $\SS$:

the infimum of $\TT$ necessarily admitted by $\TT$ is $\bigcap \TT$.

Proof

From Magma is Submagma of Itself:

$\struct {A, \odot} \in \SS$

Let $\TT$ be a non-empty subset of $\SS$.

From Intersection of Submagmas is Largest Submagma:

$\bigcap \TT \in \SS$

Hence, from Set of Subsets which contains Set and Intersection of Subsets is Complete Lattice:

$\struct {\SS, \subseteq}$ is a complete lattice

where $\bigcap \TT$ is the infimum of $\TT$.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.11 \ \text{(c) (1)}$